class 12

Math

Calculus

Differential Equations

Let $f:[1,∞]$ be a differentiable function such that $f(1)=2.$ If $6∫_{1}f(t)dt=3xf(x)−x_{3}$ for all $x≥1,$ then the value of $f(2)$ is

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Form the differential equation of the family of hyperbola having foci on x-axis and center at the origin.

The general solution of a differential equation of the type $dydx +P_{1}x=Q_{1}$is(A) $ye_{∫P_{1}dy}=∫(Q_{1}e_{∫P_{1}dy})dy+C$ (B) $ye˙_{∫P_{1}dx}=∫(Q_{1}e_{∫P_{1}dx})dx+C$(C) $xe_{∫P_{1}dy}=∫(Q_{1}e_{∫P_{1}dy})dy+C$ (D) $xe_{∫p_{1}dx}=∫Q_{1}e_{∫p_{1}dx}dx+C$

Find the equation of a curve passing through the point (0, 1). If the slope of the tangent to the curve at any point (x, y) is equal to the sum of the x coordinate (abscissa) and the product of the x coordinate and y coordinate (ordinate) of that point.

Show that the given differential equation is homogeneous and solve each of them.$(x_{2}−y_{2})dx+2xydy=0$

In a bank, principal increases continuously at the rate of 5% per year. An amountof Rs 1000 is deposited with this bank, how much will it worth after 10 years$(e_{0.5}=1.648)$

The number of arbitrary constants in the general solution of a differential equationof fourth order are:(A) 0 (B) 2 (C) 3 (D) 4

Solve the differential equation $(tan_{−1}y−x)dy=(1+y_{2})dx$.

Show that the differential equation $(x−y)dxdy =x+2y$is homogeneous and solve it.